Properties

Label 160.2.a.c.1.1
Level $160$
Weight $2$
Character 160.1
Self dual yes
Analytic conductor $1.278$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.27760643234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 160.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843 q^{3} +1.00000 q^{5} +2.82843 q^{7} +5.00000 q^{9} +O(q^{10})\) \(q-2.82843 q^{3} +1.00000 q^{5} +2.82843 q^{7} +5.00000 q^{9} +5.65685 q^{11} -2.00000 q^{13} -2.82843 q^{15} +2.00000 q^{17} -8.00000 q^{21} -2.82843 q^{23} +1.00000 q^{25} -5.65685 q^{27} +6.00000 q^{29} -5.65685 q^{31} -16.0000 q^{33} +2.82843 q^{35} -10.0000 q^{37} +5.65685 q^{39} +2.00000 q^{41} +8.48528 q^{43} +5.00000 q^{45} +2.82843 q^{47} +1.00000 q^{49} -5.65685 q^{51} +6.00000 q^{53} +5.65685 q^{55} -11.3137 q^{59} -2.00000 q^{61} +14.1421 q^{63} -2.00000 q^{65} +2.82843 q^{67} +8.00000 q^{69} -5.65685 q^{71} -6.00000 q^{73} -2.82843 q^{75} +16.0000 q^{77} -11.3137 q^{79} +1.00000 q^{81} -2.82843 q^{83} +2.00000 q^{85} -16.9706 q^{87} +10.0000 q^{89} -5.65685 q^{91} +16.0000 q^{93} +2.00000 q^{97} +28.2843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 10 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{5} + 10 q^{9} - 4 q^{13} + 4 q^{17} - 16 q^{21} + 2 q^{25} + 12 q^{29} - 32 q^{33} - 20 q^{37} + 4 q^{41} + 10 q^{45} + 2 q^{49} + 12 q^{53} - 4 q^{61} - 4 q^{65} + 16 q^{69} - 12 q^{73} + 32 q^{77} + 2 q^{81} + 4 q^{85} + 20 q^{89} + 32 q^{93} + 4 q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.82843 −1.63299 −0.816497 0.577350i \(-0.804087\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 0 0
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) 5.65685 1.70561 0.852803 0.522233i \(-0.174901\pi\)
0.852803 + 0.522233i \(0.174901\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.82843 −0.730297
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −8.00000 −1.74574
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 0 0
\(33\) −16.0000 −2.78524
\(34\) 0 0
\(35\) 2.82843 0.478091
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 5.65685 0.905822
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 8.48528 1.29399 0.646997 0.762493i \(-0.276025\pi\)
0.646997 + 0.762493i \(0.276025\pi\)
\(44\) 0 0
\(45\) 5.00000 0.745356
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.65685 −0.792118
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 5.65685 0.762770
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.3137 −1.47292 −0.736460 0.676481i \(-0.763504\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 14.1421 1.78174
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 2.82843 0.345547 0.172774 0.984962i \(-0.444727\pi\)
0.172774 + 0.984962i \(0.444727\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) −2.82843 −0.326599
\(76\) 0 0
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) −11.3137 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.82843 −0.310460 −0.155230 0.987878i \(-0.549612\pi\)
−0.155230 + 0.987878i \(0.549612\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) −16.9706 −1.81944
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −5.65685 −0.592999
\(92\) 0 0
\(93\) 16.0000 1.65912
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 28.2843 2.84268
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −14.1421 −1.39347 −0.696733 0.717331i \(-0.745364\pi\)
−0.696733 + 0.717331i \(0.745364\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 0 0
\(107\) 14.1421 1.36717 0.683586 0.729870i \(-0.260419\pi\)
0.683586 + 0.729870i \(0.260419\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 28.2843 2.68462
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −2.82843 −0.263752
\(116\) 0 0
\(117\) −10.0000 −0.924500
\(118\) 0 0
\(119\) 5.65685 0.518563
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) 0 0
\(123\) −5.65685 −0.510061
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.82843 0.250982 0.125491 0.992095i \(-0.459949\pi\)
0.125491 + 0.992095i \(0.459949\pi\)
\(128\) 0 0
\(129\) −24.0000 −2.11308
\(130\) 0 0
\(131\) −5.65685 −0.494242 −0.247121 0.968985i \(-0.579484\pi\)
−0.247121 + 0.968985i \(0.579484\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −5.65685 −0.486864
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −11.3137 −0.959616 −0.479808 0.877373i \(-0.659294\pi\)
−0.479808 + 0.877373i \(0.659294\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) −11.3137 −0.946100
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) −2.82843 −0.233285
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 16.9706 1.38104 0.690522 0.723311i \(-0.257381\pi\)
0.690522 + 0.723311i \(0.257381\pi\)
\(152\) 0 0
\(153\) 10.0000 0.808452
\(154\) 0 0
\(155\) −5.65685 −0.454369
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) −16.9706 −1.34585
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) −14.1421 −1.10770 −0.553849 0.832617i \(-0.686841\pi\)
−0.553849 + 0.832617i \(0.686841\pi\)
\(164\) 0 0
\(165\) −16.0000 −1.24560
\(166\) 0 0
\(167\) 14.1421 1.09435 0.547176 0.837018i \(-0.315703\pi\)
0.547176 + 0.837018i \(0.315703\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 2.82843 0.213809
\(176\) 0 0
\(177\) 32.0000 2.40527
\(178\) 0 0
\(179\) −11.3137 −0.845626 −0.422813 0.906217i \(-0.638957\pi\)
−0.422813 + 0.906217i \(0.638957\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 5.65685 0.418167
\(184\) 0 0
\(185\) −10.0000 −0.735215
\(186\) 0 0
\(187\) 11.3137 0.827340
\(188\) 0 0
\(189\) −16.0000 −1.16383
\(190\) 0 0
\(191\) −16.9706 −1.22795 −0.613973 0.789327i \(-0.710430\pi\)
−0.613973 + 0.789327i \(0.710430\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) 5.65685 0.405096
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 22.6274 1.60402 0.802008 0.597314i \(-0.203765\pi\)
0.802008 + 0.597314i \(0.203765\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 16.9706 1.19110
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) −14.1421 −0.982946
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.9706 1.16830 0.584151 0.811645i \(-0.301428\pi\)
0.584151 + 0.811645i \(0.301428\pi\)
\(212\) 0 0
\(213\) 16.0000 1.09630
\(214\) 0 0
\(215\) 8.48528 0.578691
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 0 0
\(219\) 16.9706 1.14676
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 8.48528 0.568216 0.284108 0.958792i \(-0.408302\pi\)
0.284108 + 0.958792i \(0.408302\pi\)
\(224\) 0 0
\(225\) 5.00000 0.333333
\(226\) 0 0
\(227\) −19.7990 −1.31411 −0.657053 0.753845i \(-0.728197\pi\)
−0.657053 + 0.753845i \(0.728197\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −45.2548 −2.97755
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 2.82843 0.184506
\(236\) 0 0
\(237\) 32.0000 2.07862
\(238\) 0 0
\(239\) 11.3137 0.731823 0.365911 0.930650i \(-0.380757\pi\)
0.365911 + 0.930650i \(0.380757\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 14.1421 0.907218
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) 5.65685 0.357057 0.178529 0.983935i \(-0.442866\pi\)
0.178529 + 0.983935i \(0.442866\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) 0 0
\(255\) −5.65685 −0.354246
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −28.2843 −1.75750
\(260\) 0 0
\(261\) 30.0000 1.85695
\(262\) 0 0
\(263\) 19.7990 1.22086 0.610429 0.792071i \(-0.290997\pi\)
0.610429 + 0.792071i \(0.290997\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −28.2843 −1.73097
\(268\) 0 0
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −16.9706 −1.03089 −0.515444 0.856923i \(-0.672373\pi\)
−0.515444 + 0.856923i \(0.672373\pi\)
\(272\) 0 0
\(273\) 16.0000 0.968364
\(274\) 0 0
\(275\) 5.65685 0.341121
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −28.2843 −1.69334
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 8.48528 0.504398 0.252199 0.967675i \(-0.418846\pi\)
0.252199 + 0.967675i \(0.418846\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.65685 0.333914
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −5.65685 −0.331611
\(292\) 0 0
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 0 0
\(295\) −11.3137 −0.658710
\(296\) 0 0
\(297\) −32.0000 −1.85683
\(298\) 0 0
\(299\) 5.65685 0.327144
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) 5.65685 0.324978
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 2.82843 0.161427 0.0807134 0.996737i \(-0.474280\pi\)
0.0807134 + 0.996737i \(0.474280\pi\)
\(308\) 0 0
\(309\) 40.0000 2.27552
\(310\) 0 0
\(311\) 28.2843 1.60385 0.801927 0.597422i \(-0.203808\pi\)
0.801927 + 0.597422i \(0.203808\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 14.1421 0.796819
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 33.9411 1.90034
\(320\) 0 0
\(321\) −40.0000 −2.23258
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 50.9117 2.81542
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −5.65685 −0.310929 −0.155464 0.987841i \(-0.549687\pi\)
−0.155464 + 0.987841i \(0.549687\pi\)
\(332\) 0 0
\(333\) −50.0000 −2.73998
\(334\) 0 0
\(335\) 2.82843 0.154533
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) −5.65685 −0.307238
\(340\) 0 0
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 8.00000 0.430706
\(346\) 0 0
\(347\) −8.48528 −0.455514 −0.227757 0.973718i \(-0.573139\pi\)
−0.227757 + 0.973718i \(0.573139\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 11.3137 0.603881
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) −5.65685 −0.300235
\(356\) 0 0
\(357\) −16.0000 −0.846810
\(358\) 0 0
\(359\) −22.6274 −1.19423 −0.597115 0.802156i \(-0.703686\pi\)
−0.597115 + 0.802156i \(0.703686\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −59.3970 −3.11753
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −8.48528 −0.442928 −0.221464 0.975169i \(-0.571084\pi\)
−0.221464 + 0.975169i \(0.571084\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 16.9706 0.881068
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) −2.82843 −0.146059
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 22.6274 1.16229 0.581146 0.813799i \(-0.302604\pi\)
0.581146 + 0.813799i \(0.302604\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) −36.7696 −1.87884 −0.939418 0.342773i \(-0.888634\pi\)
−0.939418 + 0.342773i \(0.888634\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 42.4264 2.15666
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −5.65685 −0.286079
\(392\) 0 0
\(393\) 16.0000 0.807093
\(394\) 0 0
\(395\) −11.3137 −0.569254
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 11.3137 0.563576
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −56.5685 −2.80400
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 16.9706 0.837096
\(412\) 0 0
\(413\) −32.0000 −1.57462
\(414\) 0 0
\(415\) −2.82843 −0.138842
\(416\) 0 0
\(417\) 32.0000 1.56705
\(418\) 0 0
\(419\) 11.3137 0.552711 0.276355 0.961056i \(-0.410873\pi\)
0.276355 + 0.961056i \(0.410873\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 0 0
\(423\) 14.1421 0.687614
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −5.65685 −0.273754
\(428\) 0 0
\(429\) 32.0000 1.54497
\(430\) 0 0
\(431\) 5.65685 0.272481 0.136241 0.990676i \(-0.456498\pi\)
0.136241 + 0.990676i \(0.456498\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −16.9706 −0.813676
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −22.6274 −1.07995 −0.539974 0.841682i \(-0.681566\pi\)
−0.539974 + 0.841682i \(0.681566\pi\)
\(440\) 0 0
\(441\) 5.00000 0.238095
\(442\) 0 0
\(443\) −2.82843 −0.134383 −0.0671913 0.997740i \(-0.521404\pi\)
−0.0671913 + 0.997740i \(0.521404\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) 28.2843 1.33780
\(448\) 0 0
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) 11.3137 0.532742
\(452\) 0 0
\(453\) −48.0000 −2.25524
\(454\) 0 0
\(455\) −5.65685 −0.265197
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) −11.3137 −0.528079
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 8.48528 0.394344 0.197172 0.980369i \(-0.436824\pi\)
0.197172 + 0.980369i \(0.436824\pi\)
\(464\) 0 0
\(465\) 16.0000 0.741982
\(466\) 0 0
\(467\) 14.1421 0.654420 0.327210 0.944952i \(-0.393892\pi\)
0.327210 + 0.944952i \(0.393892\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 50.9117 2.34589
\(472\) 0 0
\(473\) 48.0000 2.20704
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 30.0000 1.37361
\(478\) 0 0
\(479\) 33.9411 1.55081 0.775405 0.631464i \(-0.217546\pi\)
0.775405 + 0.631464i \(0.217546\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) 22.6274 1.02958
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) −31.1127 −1.40985 −0.704925 0.709281i \(-0.749020\pi\)
−0.704925 + 0.709281i \(0.749020\pi\)
\(488\) 0 0
\(489\) 40.0000 1.80886
\(490\) 0 0
\(491\) −39.5980 −1.78703 −0.893516 0.449032i \(-0.851769\pi\)
−0.893516 + 0.449032i \(0.851769\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 28.2843 1.27128
\(496\) 0 0
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) −40.0000 −1.78707
\(502\) 0 0
\(503\) 8.48528 0.378340 0.189170 0.981944i \(-0.439420\pi\)
0.189170 + 0.981944i \(0.439420\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) 25.4558 1.13053
\(508\) 0 0
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) −16.9706 −0.750733
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −14.1421 −0.623177
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 5.65685 0.248308
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) 8.48528 0.371035 0.185518 0.982641i \(-0.440604\pi\)
0.185518 + 0.982641i \(0.440604\pi\)
\(524\) 0 0
\(525\) −8.00000 −0.349149
\(526\) 0 0
\(527\) −11.3137 −0.492833
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) −56.5685 −2.45487
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 14.1421 0.611418
\(536\) 0 0
\(537\) 32.0000 1.38090
\(538\) 0 0
\(539\) 5.65685 0.243658
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) −39.5980 −1.69931
\(544\) 0 0
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) −42.4264 −1.81402 −0.907011 0.421107i \(-0.861642\pi\)
−0.907011 + 0.421107i \(0.861642\pi\)
\(548\) 0 0
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 0 0
\(555\) 28.2843 1.20060
\(556\) 0 0
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 0 0
\(559\) −16.9706 −0.717778
\(560\) 0 0
\(561\) −32.0000 −1.35104
\(562\) 0 0
\(563\) 19.7990 0.834428 0.417214 0.908808i \(-0.363007\pi\)
0.417214 + 0.908808i \(0.363007\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 2.82843 0.118783
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) −28.2843 −1.18366 −0.591830 0.806063i \(-0.701594\pi\)
−0.591830 + 0.806063i \(0.701594\pi\)
\(572\) 0 0
\(573\) 48.0000 2.00523
\(574\) 0 0
\(575\) −2.82843 −0.117954
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 0 0
\(579\) −50.9117 −2.11582
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 33.9411 1.40570
\(584\) 0 0
\(585\) −10.0000 −0.413449
\(586\) 0 0
\(587\) 25.4558 1.05068 0.525338 0.850894i \(-0.323939\pi\)
0.525338 + 0.850894i \(0.323939\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −16.9706 −0.698076
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 5.65685 0.231908
\(596\) 0 0
\(597\) −64.0000 −2.61935
\(598\) 0 0
\(599\) −11.3137 −0.462266 −0.231133 0.972922i \(-0.574243\pi\)
−0.231133 + 0.972922i \(0.574243\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 14.1421 0.575912
\(604\) 0 0
\(605\) 21.0000 0.853771
\(606\) 0 0
\(607\) 2.82843 0.114802 0.0574012 0.998351i \(-0.481719\pi\)
0.0574012 + 0.998351i \(0.481719\pi\)
\(608\) 0 0
\(609\) −48.0000 −1.94506
\(610\) 0 0
\(611\) −5.65685 −0.228852
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) −5.65685 −0.228106
\(616\) 0 0
\(617\) 10.0000 0.402585 0.201292 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) 0 0
\(619\) 45.2548 1.81895 0.909473 0.415764i \(-0.136486\pi\)
0.909473 + 0.415764i \(0.136486\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 0 0
\(623\) 28.2843 1.13319
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) 16.9706 0.675587 0.337794 0.941220i \(-0.390319\pi\)
0.337794 + 0.941220i \(0.390319\pi\)
\(632\) 0 0
\(633\) −48.0000 −1.90783
\(634\) 0 0
\(635\) 2.82843 0.112243
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −28.2843 −1.11891
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 31.1127 1.22697 0.613483 0.789708i \(-0.289768\pi\)
0.613483 + 0.789708i \(0.289768\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) 14.1421 0.555985 0.277992 0.960583i \(-0.410331\pi\)
0.277992 + 0.960583i \(0.410331\pi\)
\(648\) 0 0
\(649\) −64.0000 −2.51222
\(650\) 0 0
\(651\) 45.2548 1.77368
\(652\) 0 0
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) −5.65685 −0.221032
\(656\) 0 0
\(657\) −30.0000 −1.17041
\(658\) 0 0
\(659\) 33.9411 1.32216 0.661079 0.750316i \(-0.270099\pi\)
0.661079 + 0.750316i \(0.270099\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 11.3137 0.439388
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −16.9706 −0.657103
\(668\) 0 0
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) −11.3137 −0.436761
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) −5.65685 −0.217732
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) 5.65685 0.217090
\(680\) 0 0
\(681\) 56.0000 2.14592
\(682\) 0 0
\(683\) −14.1421 −0.541134 −0.270567 0.962701i \(-0.587211\pi\)
−0.270567 + 0.962701i \(0.587211\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) −39.5980 −1.51076
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 28.2843 1.07598 0.537992 0.842950i \(-0.319183\pi\)
0.537992 + 0.842950i \(0.319183\pi\)
\(692\) 0 0
\(693\) 80.0000 3.03895
\(694\) 0 0
\(695\) −11.3137 −0.429153
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −28.2843 −1.06981
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) −5.65685 −0.212748
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) −56.5685 −2.12149
\(712\) 0 0
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) −11.3137 −0.423109
\(716\) 0 0
\(717\) −32.0000 −1.19506
\(718\) 0 0
\(719\) 11.3137 0.421930 0.210965 0.977494i \(-0.432339\pi\)
0.210965 + 0.977494i \(0.432339\pi\)
\(720\) 0 0
\(721\) −40.0000 −1.48968
\(722\) 0 0
\(723\) −73.5391 −2.73495
\(724\) 0 0
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 25.4558 0.944105 0.472052 0.881570i \(-0.343513\pi\)
0.472052 + 0.881570i \(0.343513\pi\)
\(728\) 0 0
\(729\) −43.0000 −1.59259
\(730\) 0 0
\(731\) 16.9706 0.627679
\(732\) 0 0
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) 0 0
\(735\) −2.82843 −0.104328
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 11.3137 0.416181 0.208091 0.978110i \(-0.433275\pi\)
0.208091 + 0.978110i \(0.433275\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −14.1421 −0.518825 −0.259412 0.965767i \(-0.583529\pi\)
−0.259412 + 0.965767i \(0.583529\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 0 0
\(747\) −14.1421 −0.517434
\(748\) 0 0
\(749\) 40.0000 1.46157
\(750\) 0 0
\(751\) 16.9706 0.619265 0.309632 0.950856i \(-0.399794\pi\)
0.309632 + 0.950856i \(0.399794\pi\)
\(752\) 0 0
\(753\) −16.0000 −0.583072
\(754\) 0 0
\(755\) 16.9706 0.617622
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 45.2548 1.64265
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) −50.9117 −1.84313
\(764\) 0 0
\(765\) 10.0000 0.361551
\(766\) 0 0
\(767\) 22.6274 0.817029
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 39.5980 1.42609
\(772\) 0 0
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) −5.65685 −0.203200
\(776\) 0 0
\(777\) 80.0000 2.86998
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) −33.9411 −1.21296
\(784\) 0 0
\(785\) −18.0000 −0.642448
\(786\) 0 0
\(787\) −8.48528 −0.302468 −0.151234 0.988498i \(-0.548325\pi\)
−0.151234 + 0.988498i \(0.548325\pi\)
\(788\) 0 0
\(789\) −56.0000 −1.99365
\(790\) 0 0
\(791\) 5.65685 0.201135
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) −16.9706 −0.601884
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 5.65685 0.200125
\(800\) 0 0
\(801\) 50.0000 1.76666
\(802\) 0 0
\(803\) −33.9411 −1.19776
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) 50.9117 1.79218
\(808\) 0 0
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) 5.65685 0.198639 0.0993195 0.995056i \(-0.468333\pi\)
0.0993195 + 0.995056i \(0.468333\pi\)
\(812\) 0 0
\(813\) 48.0000 1.68343
\(814\) 0 0
\(815\) −14.1421 −0.495377
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −28.2843 −0.988332
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) −25.4558 −0.887335 −0.443667 0.896191i \(-0.646323\pi\)
−0.443667 + 0.896191i \(0.646323\pi\)
\(824\) 0 0
\(825\) −16.0000 −0.557048
\(826\) 0 0
\(827\) −31.1127 −1.08189 −0.540947 0.841057i \(-0.681934\pi\)
−0.540947 + 0.841057i \(0.681934\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 28.2843 0.981170
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 14.1421 0.489409
\(836\) 0 0
\(837\) 32.0000 1.10608
\(838\) 0 0
\(839\) −11.3137 −0.390593 −0.195296 0.980744i \(-0.562567\pi\)
−0.195296 + 0.980744i \(0.562567\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 84.8528 2.92249
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 59.3970 2.04090
\(848\) 0 0
\(849\) −24.0000 −0.823678
\(850\) 0 0
\(851\) 28.2843 0.969572
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) −33.9411 −1.15806 −0.579028 0.815308i \(-0.696568\pi\)
−0.579028 + 0.815308i \(0.696568\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) 0 0
\(863\) 42.4264 1.44421 0.722106 0.691783i \(-0.243174\pi\)
0.722106 + 0.691783i \(0.243174\pi\)
\(864\) 0 0
\(865\) −2.00000 −0.0680020
\(866\) 0 0
\(867\) 36.7696 1.24876
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) 0 0
\(871\) −5.65685 −0.191675
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 2.82843 0.0956183
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 0 0
\(879\) 28.2843 0.954005
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) −2.82843 −0.0951842 −0.0475921 0.998867i \(-0.515155\pi\)
−0.0475921 + 0.998867i \(0.515155\pi\)
\(884\) 0 0
\(885\) 32.0000 1.07567
\(886\) 0 0
\(887\) 2.82843 0.0949693 0.0474846 0.998872i \(-0.484879\pi\)
0.0474846 + 0.998872i \(0.484879\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 5.65685 0.189512
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −11.3137 −0.378176
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) −33.9411 −1.13200
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) −67.8823 −2.25898
\(904\) 0 0
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) 25.4558 0.845247 0.422624 0.906305i \(-0.361109\pi\)
0.422624 + 0.906305i \(0.361109\pi\)
\(908\) 0 0
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −5.65685 −0.187420 −0.0937100 0.995600i \(-0.529873\pi\)
−0.0937100 + 0.995600i \(0.529873\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) 5.65685 0.187010
\(916\) 0 0
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) −33.9411 −1.11961 −0.559807 0.828623i \(-0.689125\pi\)
−0.559807 + 0.828623i \(0.689125\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 0 0
\(923\) 11.3137 0.372395
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 0 0
\(927\) −70.7107 −2.32244
\(928\) 0 0
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −80.0000 −2.61908
\(934\) 0 0
\(935\) 11.3137 0.369998
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 16.9706 0.553813
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 0 0
\(943\) −5.65685 −0.184213
\(944\) 0 0
\(945\) −16.0000 −0.520480
\(946\) 0 0
\(947\) −42.4264 −1.37867 −0.689336 0.724441i \(-0.742098\pi\)
−0.689336 + 0.724441i \(0.742098\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) −84.8528 −2.75154
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −16.9706 −0.549155
\(956\) 0 0
\(957\) −96.0000 −3.10324
\(958\) 0 0
\(959\) −16.9706 −0.548008
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 70.7107 2.27862
\(964\) 0 0
\(965\) 18.0000 0.579441
\(966\) 0 0
\(967\) −42.4264 −1.36434 −0.682171 0.731193i \(-0.738964\pi\)
−0.682171 + 0.731193i \(0.738964\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −28.2843 −0.907685 −0.453843 0.891082i \(-0.649947\pi\)
−0.453843 + 0.891082i \(0.649947\pi\)
\(972\) 0 0
\(973\) −32.0000 −1.02587
\(974\) 0 0
\(975\) 5.65685 0.181164
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 56.5685 1.80794
\(980\) 0 0
\(981\) −90.0000 −2.87348
\(982\) 0 0
\(983\) −2.82843 −0.0902128 −0.0451064 0.998982i \(-0.514363\pi\)
−0.0451064 + 0.998982i \(0.514363\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) −22.6274 −0.720239
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 16.9706 0.539088 0.269544 0.962988i \(-0.413127\pi\)
0.269544 + 0.962988i \(0.413127\pi\)
\(992\) 0 0
\(993\) 16.0000 0.507745
\(994\) 0 0
\(995\) 22.6274 0.717337
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 0 0
\(999\) 56.5685 1.78975
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 160.2.a.c.1.1 2
3.2 odd 2 1440.2.a.o.1.2 2
4.3 odd 2 inner 160.2.a.c.1.2 yes 2
5.2 odd 4 800.2.c.f.449.4 4
5.3 odd 4 800.2.c.f.449.2 4
5.4 even 2 800.2.a.m.1.2 2
7.6 odd 2 7840.2.a.bf.1.2 2
8.3 odd 2 320.2.a.g.1.1 2
8.5 even 2 320.2.a.g.1.2 2
12.11 even 2 1440.2.a.o.1.1 2
15.2 even 4 7200.2.f.bh.6049.3 4
15.8 even 4 7200.2.f.bh.6049.1 4
15.14 odd 2 7200.2.a.cm.1.1 2
16.3 odd 4 1280.2.d.l.641.3 4
16.5 even 4 1280.2.d.l.641.4 4
16.11 odd 4 1280.2.d.l.641.2 4
16.13 even 4 1280.2.d.l.641.1 4
20.3 even 4 800.2.c.f.449.3 4
20.7 even 4 800.2.c.f.449.1 4
20.19 odd 2 800.2.a.m.1.1 2
24.5 odd 2 2880.2.a.bk.1.2 2
24.11 even 2 2880.2.a.bk.1.1 2
28.27 even 2 7840.2.a.bf.1.1 2
40.3 even 4 1600.2.c.n.449.2 4
40.13 odd 4 1600.2.c.n.449.3 4
40.19 odd 2 1600.2.a.bc.1.2 2
40.27 even 4 1600.2.c.n.449.4 4
40.29 even 2 1600.2.a.bc.1.1 2
40.37 odd 4 1600.2.c.n.449.1 4
60.23 odd 4 7200.2.f.bh.6049.4 4
60.47 odd 4 7200.2.f.bh.6049.2 4
60.59 even 2 7200.2.a.cm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.a.c.1.1 2 1.1 even 1 trivial
160.2.a.c.1.2 yes 2 4.3 odd 2 inner
320.2.a.g.1.1 2 8.3 odd 2
320.2.a.g.1.2 2 8.5 even 2
800.2.a.m.1.1 2 20.19 odd 2
800.2.a.m.1.2 2 5.4 even 2
800.2.c.f.449.1 4 20.7 even 4
800.2.c.f.449.2 4 5.3 odd 4
800.2.c.f.449.3 4 20.3 even 4
800.2.c.f.449.4 4 5.2 odd 4
1280.2.d.l.641.1 4 16.13 even 4
1280.2.d.l.641.2 4 16.11 odd 4
1280.2.d.l.641.3 4 16.3 odd 4
1280.2.d.l.641.4 4 16.5 even 4
1440.2.a.o.1.1 2 12.11 even 2
1440.2.a.o.1.2 2 3.2 odd 2
1600.2.a.bc.1.1 2 40.29 even 2
1600.2.a.bc.1.2 2 40.19 odd 2
1600.2.c.n.449.1 4 40.37 odd 4
1600.2.c.n.449.2 4 40.3 even 4
1600.2.c.n.449.3 4 40.13 odd 4
1600.2.c.n.449.4 4 40.27 even 4
2880.2.a.bk.1.1 2 24.11 even 2
2880.2.a.bk.1.2 2 24.5 odd 2
7200.2.a.cm.1.1 2 15.14 odd 2
7200.2.a.cm.1.2 2 60.59 even 2
7200.2.f.bh.6049.1 4 15.8 even 4
7200.2.f.bh.6049.2 4 60.47 odd 4
7200.2.f.bh.6049.3 4 15.2 even 4
7200.2.f.bh.6049.4 4 60.23 odd 4
7840.2.a.bf.1.1 2 28.27 even 2
7840.2.a.bf.1.2 2 7.6 odd 2